Integrand size = 22, antiderivative size = 171 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=-\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}+\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {101, 156, 157, 12, 95, 214} \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\frac {\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{7/2}}-\frac {d \sqrt {a+b x} (b c-15 a d)}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} (b c-5 a d)}{4 a c^2 x \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}} \]
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 157
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}+\frac {\int \frac {\frac {1}{2} (b c-5 a d)-2 b d x}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{2 c} \\ & = -\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}-\frac {\int \frac {\frac {1}{4} \left (b^2 c^2+6 a b c d-15 a^2 d^2\right )+\frac {1}{2} b d (b c-5 a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{2 a c^2} \\ & = -\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}+\frac {\int -\frac {(b c-a d) \left (b^2 c^2+6 a b c d-15 a^2 d^2\right )}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a c^3 (b c-a d)} \\ & = -\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}-\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a c^3} \\ & = -\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}-\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a c^3} \\ & = -\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}+\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{7/2}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (-b c x (c+d x)+a \left (-2 c^2+5 c d x+15 d^2 x^2\right )\right )}{4 a c^3 x^2 \sqrt {c+d x}}+\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(466\) vs. \(2(139)=278\).
Time = 0.53 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.73
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{3} x^{3}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c \,d^{2} x^{3}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{2} x^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d \,x^{2}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} x^{2}-30 a \,d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 b c d \,x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-10 a c d x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 b \,c^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 a \,c^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{8 a \,c^{3} \sqrt {a c}\, x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}}\) | \(467\) |
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Time = 0.56 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.77 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\left [-\frac {{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - 5 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a^{2} c^{4} d x^{3} + a^{2} c^{5} x^{2}\right )}}, -\frac {{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - 5 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a^{2} c^{4} d x^{3} + a^{2} c^{5} x^{2}\right )}}\right ] \]
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\[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a + b x}}{x^{3} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1092 vs. \(2 (139) = 278\).
Time = 1.61 (sec) , antiderivative size = 1092, normalized size of antiderivative = 6.39 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} b^{2} d^{2}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c^{3} {\left | b \right |}} + \frac {{\left (\sqrt {b d} b^{4} c^{2} + 6 \, \sqrt {b d} a b^{3} c d - 15 \, \sqrt {b d} a^{2} b^{2} d^{2}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{4 \, \sqrt {-a b c d} a b c^{3} {\left | b \right |}} - \frac {\sqrt {b d} b^{10} c^{5} - 11 \, \sqrt {b d} a b^{9} c^{4} d + 34 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} - 46 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} + 29 \, \sqrt {b d} a^{4} b^{6} c d^{4} - 7 \, \sqrt {b d} a^{5} b^{5} d^{5} - 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{8} c^{4} + 28 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{7} c^{3} d - 26 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{6} c^{2} d^{2} - 20 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{5} c d^{3} + 21 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{4} d^{4} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{6} c^{3} - 19 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{5} c^{2} d - 11 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{4} c d^{2} - 21 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{3} d^{3} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{4} c^{2} + 2 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{3} c d + 7 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b^{2} d^{2}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2} a c^{3} {\left | b \right |}} \]
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Timed out. \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x^3\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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